Abstract
A positive and normalised real linear functional on the set of bounded continuous functions can be characterised as the integral of a σ-additive probability measure, by the F. Riesz Representation Theorem. In this paper, we look at the finitely additive extensions of such a functional to the set of all bounded random variables, and prove that they are determined by Riesz’ extension to lower semi-continuous functions. In doing so, we establish links with Daniell’s approach to integration, Walley’s theory of coherent lower previsions, and de Finetti’s Representation Theorem for exchangeable random variables.
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de Cooman, G., Miranda, E. (2008). The F. Riesz Representation Theorem and Finite Additivity. In: Dubois, D., Lubiano, M.A., Prade, H., Gil, M.Á., Grzegorzewski, P., Hryniewicz, O. (eds) Soft Methods for Handling Variability and Imprecision. Advances in Soft Computing, vol 48. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85027-4_30
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DOI: https://doi.org/10.1007/978-3-540-85027-4_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-85026-7
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